pentane Li
THF Li
THF
Li
vinyllithium
Et2O LiLi
+ LiBr
+ LiCl + LiBr
+ LiCl
You will notice secondary alkyllithiums, an aryllithium, and two vinyllithiums. The only other functional
groups are alkenes and an ether. So far, that is quite like the formation
X
Mg
R
R
Mg
X
R
Mg
X
RX
R
Mg
X
Mg
How to make organolithium reagents
Organolithium compounds may be made by a similar oxidative insertion reaction from lith- ium metal
and alkyl halides. Each inserting reaction requires two atoms of lithium and gener- ate
value, streamlines cannot cross (the flow cannot go in more than one
direction at the same time).1 To visualize a streamline in a flow, imagine
the motion of a small marked particle of fluid. For instance, we could
mark a drop of water with fluorescent dy
and so this relationship holds for flows with variable density. When the
density is constant, however, we can integrate this relationship along
the streamline to obtain p + 1 2 V 2 + gz = constant which is Bernoullis
equation for steady, constant density
makes the velocity at the wall zero, so that the rate of work by viscous
forces is also zero. At an inlet or outlet, the flow is usually normal to the
surface, and then the shear work is again zero. The shear work is rarely
important for large control vol
along a streamline can be relaxed in an irrotational flow field, which is
defined as a region where = ~ V = 0 (see Chapter 6). In that case,
Bernoullis equation can be used across streamlines. Also, in practice,
the fluid has a constant density statement
1A1V 2 1 (3.13) The solution for FD is somewhat complicated, and
without further information of the flow conditions it is difficult to tell
whether it is positive or negative. It is good practice, therefore, not to
try to anticipate the direction of FD or
area increasing velocity 4.2. BERNOULLIS EQUATION 81 Figure 4.5:
Flow in a one-dimensional duct where the inlet and outlet directions
are aligned. This flow case was first analyzed using a large control
volume in Section 3.5.2. With V2 > V1, p2 < p1 incre
EQUATIONS OF MOTION IN INTEGRAL FORM Figure 3.15: Flow through
a sudden contraction (left), and a sudden expansion (right). Flow is from
left to right. From Visualized Flow, Japan Society of Mechanical
Engineers, Pergamon Press, 1988. If the heat and work
curvature becomes very large, the pressure across streamlines can only
vary through hydrostatic pressure differences. If the effects of gravity
are not important, and R , then equation 4.7 gives that p/n = 0,
that is, the pressure is constant across strea
examine the forces acting on the particle that follows this path, which
leads us directly to Bernoulls equation. Bernoullis equation is obtained
by applying Newtons second law between two points on a streamline
(the complete derivation is given in Section
2A2V 2 2 sin That is, R y ext = p2gA2 sin 2A2V 2 2 sin Finally,
the resultant force by the duct on the fluid is given by Rext = R x exti + R
y extj Hence the force by the fluid on the duct is given by Rext, and
therefore FD, the force required to hold the
momentum = (2V2A2)V2 cos (1V1A1)V1 (3.14) Hence, by
Newtons second law R x ext + p1gA1 p2gA2 cos = 2A2V 2 2 cos
1A1V 2 1 and so R x ext = (p1gA1 p2gA2 cos ) 1A1V 2 1
2A2V 2 2 cos We now go through the same procedure to find the
force R y ext. The force
instantaneous velocity direction. If we mark many drops of water in this
way, all the streamlines in the flow would become visible. Particle
Image Velocimetry (PIV) is a technique to measure the instantaneous
velocity field by visualizing the motion of in
the flow approaches the cylinder, indicating that the flow is slowing
down. The central streamline stops at the cylinder surface at a point
called the stagnation point, and this particular streamline is called the
stagnation streamline (see also Figure 4.
increases with radius. At the same time, the pressure rises. However, at
the exit (B and C), the pressure must be equal to atmospheric pressure
since the streamlines at the exit are parallel. The pressure in the gap
must, therefore, be everywhere less tha
to pressure acting on area A2 = p2gA2 to the left (2) There is a force
exerted by the duct on the fluid (the external forces included in the
momentum equation are always the forces acting on the fluid). This is
intuitive, 64 CHAPTER 3. EQUATIONS OF MOTION
airfoil, the airfoil must be exerting a force on the fluid. Therefore, by
Newtons third law, there will be an equal and opposite force acting on
the airfoil. The component normal to the incoming flow contributes to
the lift, and the component in the direc
momentum flux out of control volume = resultant force acting
on fluid in control volume 3.6. MOMENTUM EQUATION 67 Figure
3.12: Fixed control volume for derivation of the integral form of the
momentum equation. We begin by considering the net momentum flux
fluid inside the control volume can be split four ways: W = W pressure
+ W grav + W viscous + W shaf t where W pressure is the rate that
work is done by forces due to pressure, W grav is the rate that work is
done against gravity, W viscous is the rate th
indirectly by using a static pressure tap located on the wall of the wind
tunnel (as shown in Figure 4.9). A differential electronic manometer
[Figure 4.9(a)] will measure the difference pm ps, and by using the
hydrostatic equation twice we obtain ps = p
forces due to external surfaces (Rext), we obtain t Z V d + Z (n V)
V dA = Z n p dA + Z g d + Fv + Rext (3.20) This is the integral form of
the momentum equation for a fixed control volume, in a
threedimensional, time-dependent flow. It is a vector equati
to be positive the net change of x-momentum during t = 2A2V 2 2
1A1V 2 1 t the net rate change of x-momentum = 2A2V 2 2 1A1V
2 1 This is the rate of increase in x-momentum experienced by the fluid
in passing through the duct. Note that the quantity AV 2
friction or flow separation. One application of the Venturi tube is as a
flow measurement device for the steady flow of a constant-density
fluid. As the fluid passes through the tube, it reaches its maximum
velocity and minimum pressure at the smallest cr
are acting. This is not the case for many practical flows. In most pipe
and duct flows, for example, the velocity gradients extend over the
entire cross-section and frictional stresses are important everywhere.
Even if the boundary layers are thin to begi
dy dz = v w dx dz = u w The procedure for finding the shape of the
streamlines is illustrated in Examples 4.1 and 4.2. Surface streamlines
are the streamlines followed by the flow very close to a solid surface.
Small, flexible tufts of yarn are sometimes
long as they are equal. Finally, for steady or unsteady constant density
flow, Z n VdA = 0 constant density (3.12) 3.5 Conservation of
Momentum We now consider the second principle of fluid motion:
conservation of momentum. We will first construct the mom
moves towards the plate. Since the flow cannot pass through the plate,
the fluid must come to rest at the point where it meets the plate. In
other words, it stagnates. The fluid along the dividing, or stagnation
streamline slows down and eventually comes
area is smaller. Therefore a fluid particle experiences an acceleration as
it moves from A to B: this part of the acceleration is described by the
convective acceleration. If the mass flow rate through the duct was also
unsteady, the local acceleration wo
(unsteady flow), it is possible for lines to cross over each other as well
as themselves. As an example, Figure 4.3(b) shows how streaklines can
be used to visualize the flow over a circular cylinder. Surface streaklines
are the streaklines followed by th